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Lunar distance method of Chauvenet
From: Paul Hirose
Date: 2020 May 25, 21:59 -0700

```In another thread I mentioned the lunar distance method of William
Chauvenet. I think it first appeared in 1851, in the Astronomical Journal:

The American Ephemeris and Nautical Almanac for the Year 1855 contains a
description in "navigator language," two examples, and tables:

https://archive.org/details/americanephemer19offigoog/page/n504/mode/2up

His manual of spherical astronomy (vol. 1) says it's "the shortest and
simplest of the approximative methods when these are rendered
sufficiently accurate by the introduction of all necessary corrections
... There are briefer methods to be found in every work on Navigation,
which will (and should) be preferred in cases where only a rude
approximation to the longitude is required.":

https://archive.org/details/manualofspherica01chauiala/page/402/mode/2up

The tables are in volume 2:

https://archive.org/details/manualofspherica02chau/page/600/mode/2up

Chauvenet gives two examples in the 1855 almanac. Since my objective was
to evaluate his "approximate method" or more precisely, my computer
implementation, I calculated new simulated observations with Lunar 4.4.
The angles are within a few arc minutes of the original examples.

I could have used those examples unchanged, but that would introduce
errors (such as the lunar ephemeris) which are not the fault of the
Chauvenet method itself.

The first is a morning Sun lunar. (Of course in practice the correct
time and longitude would not be known.)

1855-09-07 08:07 UT1 (correct time)
+10 s delta T
35.5° N 30° W (correct position)
zero height of eye
75 F (24 C), air pressure 29.1 inches Hg (985 mb)

observations:

49°25.66' Moon lower limb refracted altitude
5°18.87' Sun lower limb refracted altitude
43°52.71' observed lunar distance, near to near

Almanac data:

Moon at 8 h UT1
14.78' geocentric semidiameter
54.27' horizontal parallax
+25°18.01' declination

Sun at 8 h UT1
15.88' semidiameter
0.15' horizontal parallax
+6°15.61' declination

geocentric apparent distances (center to center)
08 h UT1 45°08.52'
09 h UT1 44°41.31'
10 h UT1 44°14.12'

(Back in the day, lunar distances were typically tabulated every three
hours. However, Chauvenet's method doesn't require that. My
implementation accepts distances at any time interval. Also, it applies
a correction for second differences, i.e., a uniform lunar distance rate
is not assumed. The method of second difference correction is explained
in the interpolation section of his spherical astronomy book.)

Chauvenet solution:

45°05.300′ geocentric distance
8h07m06s observation time

correct values:
45°05.34' distance
8m07m00s time

Chauvenet solution error = +6 s.

The other example in the 1855 almanac is an evening Fomalhaut lunar.

1855-08-30 05:40 UT1 (correct time)
+10 s delta T
55° 20' S 120° 25' W (correct position)
zero height of eye
20 F (-7 C), 31 inches Hg (1050 mb) altimeter setting

observations

6°15.75' Moon lower limb refracted altitude
52°23.15' Fomalhaut refracted altitude
46°29.09' Moon far limb to Fomalhaut observed distance

almanac data

Moon at 6 h UT1
16.40' geocentric semidiameter
1°00.19' horizontal parallax
+3°54.45' declination

Fomalhaut
-30°23.13' declination

geocentric apparent distance from Moon center:
05 h UT1 44°59.00'
06 h UT1 45°32.55'
07 h UT1 46°06.13'

Chauvenet solution:

45°21.352′ geocentric distance
5h39m59s observation time

45°21.36' correct distance
5h40m00s correct time

solution error = -1 s

The examples in the 1855 almanac appear to have been constructed to
flatter Chauvenet's lunar method vs. methods that make no correction for
nonstandard atmosphere conditions. One example is at high temperature
and low barometer, the other at low temperature and high barometer. Both
have one body at low altitude.

Finally, there's the Sun lunar in his "Manual of Spherical and Practical
Astronomy." In this case my re-calculated example attempts to match
Chauvenet's angles to a tenth minute.

1856-03-10 03:12:49 UT1 (correct time)
+10 s delta T
34°56.6' N 149°39.8' W
zero height of eye

15.6 C (60.0 F) at observer
999.0 mb (29.50″ Hg) air pressure
50.0% relative humidity

observations

52°34.0' Moon lower limb altitude
8°56.4' Sun lower limb altitude
44°37.0' Moon to Sun, near to near

almanac data at 0300 UT1

Moon
16.34' geocentric semidiameter
59.97' horizontal parallax
+14°15.18' declination

Sun
16.09' geocentric semidiameter
0.15' horizontal parallax
-4°03.32' declination

geocentric apparent distance, center to center
3 h UT1 45°40.88'
4 h UT1 46°14.58'
5 h UT1 46°48.24'

Chauvenet solution

45°48.098′ geocentric distance
3h12m51s observation time

45°48.08′ correct geocentric distance
3h12m49s correct time

+2 s time error

Chauvenet's approximate method had good accuracy in all my tests. (I
have never evaluated his rigorous method.) However, I believe the most
direct route to maximum accuracy in a computer solution is to seek the
time and place that duplicates the three observed angles. This allows
all factors that affect the topocentric angles to be applied, to the
extent that they can be calculated.
```
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